John Steel wrote:
>> I would like to add a few little comments about pragmatism vs. realism.
>> First, I consider myself both a pragmatist and a realist. When I say
> what it is I believe (e.g. there are sets, or there are measurable
> cardinals) I speak as a realist. When I describe why I believe it (the
> simplicity, scope, power, coherence with previously accepted principles,
> and independently verified predictions of the theory these assumptions are
> part of), I speak as a pragmatist. I see no inconsistency here.
If "there are sets", etc. means simply the ordinary intuition
(a very strong and very useful illusion of existance, but without
pretensions on anything more) which one has when working with
a consistent formalism then, as I understand, this is not called
"realism" and this is, of course, consistent with the coherent
pragmatism as it was described above or by Prof. Friedman (or
with rationalism of Prof. Mycielski, or with formalist view on
mathematics; I do not see any essential difference).
Realism (which seems to me a very unnatural term - the reason of
a lot of misunderstanding) means considering imaginary mathematical
objects as real, as if they existed before and independently of
any formalism or a family of interrelated formalisms (due to which
they actually arose and without which they cannot exist in our
minds).
I ask the FOMers to correct me if this is a wrong understanding
of realism. The example of discussion on SOL shows that we
sometimes do not understand one another because we may use non
coherently even technical terms.
This position contradicts to pragmatist (and/or rationalist,
formalist) point of view. Moreover the realist position seems
to me extremely doubtful as a philosophy of mathematics because
it is essentially based on beliefs, instead of mathematical
knowledge (i.e. formal deductions, constructions). Otherwise
it would be called formalism or coherent pragmatism, or the
like, and would avoid using the term `belief' or at least would
be especially careful with its using. Say, one can believe that
every natural set-theoretical statement (such as CH) will be
eventually resolved in some reasonable extension of ZFC and
all these extensions will be (practically) consistent and
adopted by mathematical community. This is sufficiently
understandable (however, doubtful for me), but needs in further
comments, clarifications, explanations "why", etc. Beliefs in
an absolute existence of imaginary and illusory objects do not
seem to me as a solid or scientific ground for any philosophy
of mathematics.
Vladimir Sazonov